The number of simple modules of the Hecke algebras of type G ( r ,1, n )

Abstract

This paper classifies the simple modules of the cyclotomic Hecke algebras of type G(r, 1, n) and the affine Hecke algebras of type A in arbitrary characteristic. We do this by first showing that the simple modules of the cyclotomic Hecke algebras are indexed by the set of "Kleshchev multipartitions".

Cyclotomic Nazarov-Wenzl Algebras

“…We remark that the irreducible representations of the Ariki-Koike algebras are indexed by the u-Kleshchev multipartitions; see [Ari01,AM00]. In the special case when u i = d i · 1 R , for 1 ≤ i ≤ r and where 0 ≤ d i < char R, Kleshchev [Kle05] has shown that the simple H r,n (u)-modules are labelled by a set of multipartitions which gives the same Kashiwara crystal as the set of u-Kleshchev multipartitions of n. Hence, in this case, the simple W r,n (u)-modules are labelled by the set {(f, λ)}, where 0 ≤ f ≤ ⌊ We close by classifying the quasi-hereditary cyclotomic Nazarov-Wenzl algebras with ω 0 = 0.…”

Irreducible Specht modules for Iwahori–Hecke algebras of type $B$

“…The definition of regular bipartitions is quite complicated in general, and depends on q and Q; the original recursive definition [AM1] is derived from the theory of crystals. In the case e = ∞, it is easy to derive a non-recursive definition.…”

Crystal bases and simple modules for Hecke algebras of type $G(p,p,n)$

“…In particular, in our H q (p, n) case, q must be a root of unity. To state the result of Ariki, we have to recall the definition of the Kleshchev multipartition (see [7]). For any p-multipartition λ, the Young diagram of λ is the set It has five removable nodes.…”

“…They give a characterization of τ in terms of ω and a bijection κ between the set of Kleshchev multipartitions and the set of FLOTW partitions (see [20,Proposition 2.9, lines 3-4 on page 14]). Note that our description of h in (3.11) is actually a statement about the crystal graph, and although [20] uses JMMO's Fock space ( [30], [16]) and we use Hayashi's Fock space ( [22], [7]), the two crystals provided by the lattice of Kleshchev multipartitions and by the lattice of FLOTW partitions respectively are isomorphic to each other. It follows that the description of h in (3.11) in the context of Kleshchev's good lattices should also be valid in the context of FLOTW's good lattices.…”

“…Note that by [7], the number # Irr H q (p, n) is explicitly known. Therefore, to get a formula for # Irr H q (p, p, n) , it suffices to derive a formula for N ( m) for each integer 1 ≤ m < p satisfying m|p.…”